Two birds with one stone: one clock for two lunar scales

What it proposes

1. A time aligned orbit around the Moon, in which the readings of an ideal clock would equal the proper time of the lunar geoid, the selenoid.
2. A known linear transformation from those readings to Lunar Coordinate Time, so that one clock would realize both candidate definitions at once.
3. Numerical simulations of how far such a clock would drift from selenoid proper time in a more realistic lunar environment.
4. The same construction for other terrestrial planets, not only for the Earth-Moon system.

This page restates what the paper says. Every clock figure on it is a model value taken from the work itself, and we neither re-derive the figures nor rank the paper against its neighbours.

The question behind the paper is a choice of definition. A lunar reference time can be taken as Lunar Coordinate Time itself, the option the authors label O1. It can instead be taken as the proper time of the lunar geoid, the selenoid, which they label O2 ^[1].

What each option costs

Yang and colleagues state the price of each. Option O1 is simple, but no clock can realize it without steering. Option O2 is convenient for users on the lunar surface, but it brings a new scaling of the spatial coordinates and of the mass parameter of the Moon ^[1].

That is the deadlock the title answers. The paper does not argue that one option is better; it looks for a clock that would satisfy both ^[1].

The time aligned orbit

The proposal is an orbit rather than a site on the surface. In a time aligned orbit, the readings of an ideal clock would equal the selenoid proper time of option O2, and a known linear transformation would convert the same readings to the coordinate time of option O1 ^[1].

The authors show that such an orbit exists. Its semi-major axis is about 1.5 lunar radius, depending slightly on the inclination ^[1]. A single clock placed there would therefore realize both definitions at once, which is where the title comes from.

How far a real clock would drift

A simulation is not an ideal clock, and the paper says so with numbers. In a more realistic lunar environment, the proper time of the simulated clock desynchronizes from selenoid proper time by up to 190 nanoseconds after a year, at a frequency offset of 6 × 10⁻¹⁵ ^[1].

The authors then give the comparison that makes those figures readable. That offset is 3.75% of the frequency difference caused in option O2 by the lunar surface topography ^[1]. Accounting for the deviation of the mean orbits in the simulations from the nominal ones, they report 13 nanoseconds and 4 × 10⁻¹⁶ instead ^[1].

For scale, the published rate between a clock on the Moon’s surface and a clock on Earth’s surface is about 56 microseconds per day ^[2]. The offsets above are much smaller, because they describe how closely one orbiting clock would track a chosen lunar reference, not how the two bodies compare.

Where it differs from the trade-off paper

Two groups take the same question in different directions, and the difference is worth naming. Defraigne, Meynadier and Bourgoin compare the options and conclude that TCL is the best option for use as the practical reference on the Moon, without defining a new scale based on a scaling of TCL ^[3]. Their argument is read in Lunar Time by Defraigne, Meynadier and Bourgoin.

Yang and colleagues do not claim that the selenoid option should win. They describe a way to realize both options with one clock, which keeps option O2 open rather than closing the choice ^[1]. What the two works show together is that the choice was still live when both were written.

What it does not settle

This is a paper about realization, not about definition. The scale the readings would be converted to is TCL, whose epoch the International Astronomical Union fixed in 2024 ^[4]Official. Nothing here says which body would operate such a clock, or when.

The arithmetic of that conversion, in runnable form, is read in the LTE440 lunar time ephemeris ^[5]. Both papers come from the same group, whose published work is set out in China and lunar time. The frame an orbiting clock would be tracked in is read in the lunar reference frame. All of our readings are listed in the papers hub.

Key numbers

Every figure below is a model value taken from the source named in its own row. Nothing in this table is our own estimate.

Status

Peer-reviewed. Astronomy & Astrophysics 707, A295, 28 December 2025^[1]. DOI 10.1051/0004-6361/202558803. arXiv 2512.23098. The full text is at Two birds with one stone: simultaneous realization of both Lunar Coordinate Time and lunar geoid time by a single orbital clock.

Why it matters

Two candidate definitions of a lunar reference time have been on the table: the coordinate time itself, and the proper time of the selenoid. This paper describes a single clock that would serve both, which turns a choice between two definitions into a question of where the clock is put.